3 Context Free Languages and Push Down Automata

1
3 - Context Free Languages and Push Down Automata

• Context Free Grammars introduction
• Context Free Grammars definitions
• Context Free Grammars normalisation
• Regular Grammars
• Push Down Automata

2
3.1 - Context-Free Grammars introduction

• The expressivity of regular languages is too
  limited for the representation of programming
  languages. Moreover, regular expressions and
  automata are not well suited to a readable
  description of languages.
• A new paradigm was introduced by Chomsky (1956),
  Schützenberger, Backus (1959) and Naur (1960)
  whereas regular languages are built recursively
  from elementary languages by means of three
  operations, a context-free language is derived
  from a set of rewriting rules which constitute
  its grammar these rules must respect some
  syntactic constraints to be a Context-Free
  Grammar (CFG).

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3.2 - Context-Free Grammars definitions

• A Context-Free Grammar is a 4-uple (N, T, S, R)
  such that
• N is a finite alphabet of non terminal symbols.
• T is a finite alphabet of terminal symbols.
• S is a particular element of N, the start
  symbol.
• R is a finite set of production rules in the form
  A ? ?, where A is a non terminal and ? is a word
  from ? (N ?T)
• The derivation relation ? between words from (N
  ?T) is defined as follows ?1A ?2 ? ?1 ? ?2 if
  A ? ? ? R. Its reflexive and transitive closure
  is written ?
• The language generated by the grammar is the set
  of words ? from T such that S ? ?. It is a
  context-free language.

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3.2 - Context-Free Grammars a linguistic
example

• The grammar G is defined by the following rules
  and its start symbol is S

NP ? jean Det ? le N ? bébé N ? berceau Vintr ?
dort Vtr ? porte Vc ? pense Prep ? dans Conj ? que
S ? NP Vintr S ? NP Vtr NP S ? NP Vc
Compl S ? S PP NP ? Det N NP ? NP PP Compl ?
Conj S PP ? Prep NP
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3.2 - Context-Free Grammars definitions

• A derivation of a word ? of the language is a
  sequence S ? ?1 ? ?2 ? ? ?. It is represented
  in a compact way by a parse tree. A parse tree is
  an ordered tree labelled with symbols from N ? T.
  The root is labelled with S and the leaves with
  terminal symbols. Every node that is not a leaf
  is labelled with a non terminal symbol A and its
  ordered daughters are labelled with symbols
  constituting a word ? such that A ? ? is a
  production rule of the grammar.
• Two grammars are equivalent if they generate the
  same language.
• A grammar is ambiguous if there exists a word
  from its language that corresponds to two
  different parse trees at least. In natural
  languages, ambiguity has an important place
  whereas in programming languages ambiguity is
  rejected.

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3.2 - Context-Free Grammars definitions

• From the grammar to the languageDetermine the
  languages generated by the following grammars.
  Are theses grammars ambiguous ? If yes, define a
  non ambiguous grammar which is equivalent to that
  one of the question.
• S ? ? aaaS
• S ? ab aSb
• S ? SS ? (S)
• S ? SS () (S) S
• S ? ? aiSai for any i such that 1 i n
• From the language to the grammarDetermine CFGs
  generating the following languages.
• L1 an bp 0 lt p lt n
• L2 an bn cm dm n, m ? ?
• L3 an bm cp n m or p m
• L4 a(ab)

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3.2 - Context-Free Grammars definitions

• Let G be the grammar E ? E E E - E (E)
  idGive all parse trees of the word id - id - id.
  Determine an equivalent non ambiguous grammar.
• If-then-else grammarsShow that the following
  grammar is ambiguous and determine a non
  ambiguous grammar which is equivalent to this
  one. S ? if B then S if B then S else S
  s B ? b Is the following grammar ambiguous ?
  S ? if B then S endif if B then S else S
  endif s B ? b

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3.3 - Context-Free Grammars normalisation

• Theorem (elimination of useless non terminal
  symbols) every CFG is equivalent to a CFG in
  which all non terminal symbols derive a word from
  T and all non terminal symbols are in a
  derivation coming from the start symbol.
• Theorem (elimination of empty production rules)
  every CFG is equivalent to a CFG in which the
  only possible empty production rule is S ? ?,
  where S is the start symbol of the grammar, and
  the other production rules do not include S in
  their right hand side.
• Theorem (Chomskys normal form) every CFG
  without empty production rule is equivalent to a
  CFG in which the production rules have one of the
  following forms A ? B C, A ? a , where A, B, C
  are non terminal symbols and a is a terminal
  symbol

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3.3 - Context-Free Grammars normalisation

• Theorem (elimination of unary rules) every CFG
  is equivalent to a CFG without unary rules (a
  unary rule is in the form A ? B).
• A CFG is recursive if it admits derivation
  relation in the form A ? ?A?. It is
  left-recursive if ? ? and it is right-recursive
  if ? ?.
• Theorem (elimination of left-recursivity) every
  CFG without empty production rule is equivalent
  to a CFG which is not left-recursive
  (right-recursive).
• Theorem (Greibachs normal form) every CFG
  without empty production rule is equivalent to a
  CFG in which the production rules have the
  following form A ? a? , where A is a non
  terminal symbol, a is a terminal symbol and ? is
  a (possibly empty) sequence of non terminal
  symbols.

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3.3 - Context-Free Grammars normalization

• Clean the following grammar from useless non
  terminal symbols S ? X f X YX ? a b c Y
  Y ?Y ? a T d ZZ ? e S W T K e fK ?
  c V W ? U a Y bU ? b d X ZT ? a TV ? a
  f
• Clean the following grammar from the empty
  productions S ? S ( S ) ?
• Put the following grammar in Chomskys normal
  form S ? a B b A A ? a a S b A A B ? b
  b S a B B

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3.3 - Context-Free Grammars normalization

• Eliminate left recursivity from the following
  grammar E ? E T TT ? T ? F FF ? ( F )
  i
• Eliminate empty production rules and left
  recursivity from the following grammar (S is the
  start symbol) S ? S a T S c dT ? T b T ?
• Put the following grammar in Greibachs normal
  form A ? B C B ? C A b C ? A B a

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3.4 - Regular Grammars

• Definition A right linear regular grammar is a
  CFG that has production rules in the form A ?
  ?B or A ? ? , where A and B are a non terminal
  symbols and ? is a (possibly empty) sequence of
  terminal symbols.
• Theorem A language is regular iff it is
  generated by a regular grammar.
• The proof of this equivalence is based on the
  identification between non terminal symbols,
  terminal symbols, production rules of a CFG on
  the one hand and states, input symbols,
  transitions of an automaton on the other hand.

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3.5 - Push Down Automata

• A Push Down Automaton is a FSA which is
  associated with a stack and the transitions of
  which depend on the symbol at the top of the
  stack, which can be modified at the same time.
• Definition a Push Down Automaton is a 7-uple
  (Q, ?,i, ?s, ?, q0, F, ?) such that
• Q is a finite set of states.
• ?i is a finite input tape alphabet of symbols.
• ?s is a finite stack alphabet of symbols.
• ? is a particular element of ?s, marking the
  bottom of the stack.
• q0 is a particular element of Q, the start state
  of the automaton.
• F is a subset of Q, the accepting states of the
  automaton.
• ? is a transition relation which associates a
  source state q1 from Q, an input tape symbol a
  from ?i ? ?, a stack top symbol b from ?s,
  with a target state q2 from Q and a push word ?
  of stack symbols. This is denoted ?(q1, a, b,
  q2, ?).

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3.5 - Push Down Automata

• The principle of a Push Down Automaton
• Initialization Initially, a word s from ?i is
  written on the tape. The other positions are
  filled with the blank symbol ?i. A pointer
  indicates the beginning of the tape.The control
  unit is in the state q0 and the stack is empty
  (it includes only the bottom symbol ?).
• Transition step A transition can occur if the
  control unit is in a state q1, if the pointer
  indicates the symbol a on the tape, if there is
  the symbol b at the top of the stack and if the
  PDA has a transition (q1, a, b, q2, ?) in its
  transition relation. In this case, the pointer of
  the tape moves to the next position. The symbol b
  at the top of the stack is replaced by the string
  ?. Finally, the control unit moves to state q2.
  Instead of a, if the transition includes the
  empty word ?, the symbol of the tape indicated by
  the pointer does not matter and the pointer does
  not move.
• Termination If in a configuration of the PDA, no
  transition is possible, the PDA stops. At this
  moment, if the unit control is in an accepting
  state, if the stack is empty and the input word
  totally read, this one is said to be accepted by
  the PDA.

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3.5 - Push Down Automata

• A Deterministic Push Down Automaton (DPDA) is a
  PDA with the following property from any state
  q1, any input tape symbol a and any stack top
  symbol b, there is one possible transition at
  most in the transition relation of the
  automaton.A PDA that is not a DPDA is a Non
  Deterministic Push Down Automaton (NPDA).
• There are languages that are recognized by NPDA
  and not by DPDA.
• Theorem a language is context-free iff it is
  recognized by a PDA.

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3.5 - Push Down Automata

• Write a recognition algorithm for a PDA.
• Let A (q0, q1, 0, 1, ?, X, ?, q0, q0,
  q1, ?) be an automaton such that ? (q0, 1,
  ?, q0, ? X), (q0, 1, X, q0, X X), (q0, 0, X, q1,
  X), (q1, 1, X , q1, ?) , (q1, 0, ?, q0, ?,)
  Describe the language recognized by this PDA
  and give a grammar that generates this language.
• Build PDA recognizing the following languages
  (DPDA if it is possible)
• anbn n 1
• anbn n 0
• apbpqcq p 0, q 1
• The language over the alphabet a, b that
  contains as many a as b.
• The language of palindromes.
• Build PDA recognizing the language generated by
  the following grammars.
• S ? if_then S if_then S else S a
• S ? S T S T U TT ? T a T b S U cU ? S T U
  T a T b